The Poisson distribution flashcards

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What is a Poisson distribution used to model?

The number of occurrences of a random, independent event within a fixed interval of time or space.

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What are the three main conditions for a Poisson distribution?

Events occur singly (one at a time), independently of each other, and at a constant average rate (λ).

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What is the parameter of a Poisson distribution and what does it represent?

The parameter is λ (lambda). It represents the mean (or average) number of occurrences in the given interval.

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What is the formula for the probability of 'r' occurrences in a Poisson distribution?

$P(X=r) = \frac{e^{-\lambda} \lambda^r}{r!}$

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What is the mean (Expected Value) of a variable $X \sim Po(\lambda)$?

$E(X) = \lambda$

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What is the variance of a variable $X \sim Po(\lambda)$?

$Var(X) = \lambda$

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What is the unique relationship between the mean and variance of a Poisson distribution?

They are equal. $E(X) = Var(X) = \lambda$. This is a key identifying feature.

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Under what conditions can a Binomial distribution $B(n, p)$ be approximated by a Poisson distribution?

When $n$ is large (e.g., $n > 50$) and $p$ is small (e.g., $p < 0.1$).

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When approximating $B(n, p)$ with $Po(\lambda)$, what value does $\lambda$ take?

$lambda$ is set to the mean of the binomial distribution: $\lambda = np$.

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A question gives an average rate of 12 events per hour. What is $\lambda$ for a 15-minute period?

The interval is 15/60 = 1/4 of an hour. So, the new $\lambda = 12 \times \frac{1}{4} = 3$.

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How do you calculate $P(X > 4)$ using cumulative tables or a calculator's CDF?

$P(X > 4) = 1 - P(X \le 4)$. Remember 'greater than' is '1 minus less than or equal to'.

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How can you calculate $P(X=5)$ using only a cumulative Poisson function?

$P(X=5) = P(X \le 5) - P(X \le 4)$.